3.141 \(\int \frac {1}{x (a^5+x^5)} \, dx\)

Optimal. Leaf size=22 \[ \frac {\log (x)}{a^5}-\frac {\log \left (a^5+x^5\right )}{5 a^5} \]

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 36, 29, 31} \[ \frac {\log (x)}{a^5}-\frac {\log \left (a^5+x^5\right )}{5 a^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a^5 + x^5)),x]

[Out]

Log[x]/a^5 - Log[a^5 + x^5]/(5*a^5)

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x \left (a^5+x^5\right )} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{x \left (a^5+x\right )} \, dx,x,x^5\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^5\right )}{5 a^5}-\frac {\operatorname {Subst}\left (\int \frac {1}{a^5+x} \, dx,x,x^5\right )}{5 a^5}\\ &=\frac {\log (x)}{a^5}-\frac {\log \left (a^5+x^5\right )}{5 a^5}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \[ \frac {\log (x)}{a^5}-\frac {\log \left (a^5+x^5\right )}{5 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a^5 + x^5)),x]

[Out]

Log[x]/a^5 - Log[a^5 + x^5]/(5*a^5)

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IntegrateAlgebraic [A]  time = 0.01, size = 22, normalized size = 1.00 \[ \frac {\log (x)}{a^5}-\frac {\log \left (a^5+x^5\right )}{5 a^5} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x*(a^5 + x^5)),x]

[Out]

Log[x]/a^5 - Log[a^5 + x^5]/(5*a^5)

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fricas [A]  time = 1.01, size = 18, normalized size = 0.82 \[ -\frac {\log \left (a^{5} + x^{5}\right ) - 5 \, \log \relax (x)}{5 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^5+x^5),x, algorithm="fricas")

[Out]

-1/5*(log(a^5 + x^5) - 5*log(x))/a^5

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giac [A]  time = 0.90, size = 22, normalized size = 1.00 \[ -\frac {\log \left ({\left | a^{5} + x^{5} \right |}\right )}{5 \, a^{5}} + \frac {\log \left ({\left | x \right |}\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^5+x^5),x, algorithm="giac")

[Out]

-1/5*log(abs(a^5 + x^5))/a^5 + log(abs(x))/a^5

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maple [A]  time = 0.31, size = 21, normalized size = 0.95

method	result	size
risch	\(\frac {\ln \relax (x )}{a^{5}}-\frac {\ln \left (a^{5}+x^{5}\right )}{5 a^{5}}\)	\(21\)
default	\(\frac {\ln \relax (x )}{a^{5}}-\frac {\ln \left (a +x \right )}{5 a^{5}}-\frac {\ln \left (a^{4}-a^{3} x +a^{2} x^{2}-a \,x^{3}+x^{4}\right )}{5 a^{5}}\)	\(49\)
norman	\(\frac {\ln \relax (x )}{a^{5}}-\frac {\ln \left (a +x \right )}{5 a^{5}}-\frac {\ln \left (a^{4}-a^{3} x +a^{2} x^{2}-a \,x^{3}+x^{4}\right )}{5 a^{5}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a^5+x^5),x,method=_RETURNVERBOSE)

[Out]

ln(x)/a^5-1/5*ln(a^5+x^5)/a^5

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maxima [A]  time = 0.43, size = 23, normalized size = 1.05 \[ -\frac {\log \left (a^{5} + x^{5}\right )}{5 \, a^{5}} + \frac {\log \left (x^{5}\right )}{5 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^5+x^5),x, algorithm="maxima")

[Out]

-1/5*log(a^5 + x^5)/a^5 + 1/5*log(x^5)/a^5

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mupad [B]  time = 0.27, size = 18, normalized size = 0.82 \[ -\frac {\ln \left (a^5+x^5\right )-5\,\ln \relax (x)}{5\,a^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a^5 + x^5)),x)

[Out]

-(log(a^5 + x^5) - 5*log(x))/(5*a^5)

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sympy [A]  time = 0.26, size = 19, normalized size = 0.86 \[ \frac {\log {\relax (x )}}{a^{5}} - \frac {\log {\left (a^{5} + x^{5} \right )}}{5 a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a**5+x**5),x)

[Out]

log(x)/a**5 - log(a**5 + x**5)/(5*a**5)

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